Status: drafted 2026-08-16, not posted. Posting is an operator action; this hunt may not open issues on its own behalf, and this file is the artifact, not the act.
Should this be an issue at all?
The repository's rule (AGENTS.md, "Observations, and the work roster") is that an observation is public and a lead is private. An issue says "this is true and unresolved"; the roster says "this one is next".
The bridge below qualifies on all three counts:
- It is a statement about the subject, not about our allocation. It says a published open conjecture already follows from an earlier published theorem. It is true or false independently of whether this laboratory ever works on it again.
- A stranger can check it. Both papers are open access, both statements are quoted verbatim in
PRIOR-ART.md, and the specialization is four hypothesis checks, none of them long. - It is unresolved in the literature. Neither paper cites the other, and the conjecture is still printed as open.
What would make it a lead instead, and therefore private, is a plan to write it up, contact the authors, or otherwise pursue it. That decision is the operator's and belongs in the roster in fulcrum, not here. The issue below states the fact and stops.
One caution worth stating in the issue itself: this is a literature observation, not a theorem of ours. The grade is "new as a literature observation, zero new mathematics", and the issue should not be written in a way that lets it read as a result of this laboratory.
Drafted issue text
Title: A published open conjecture on the prime zeta function is already a corollary of a 2022 theorem on almost periodic functions
Labels (suggested): observation, literature
Body:
While adjudicating prior art for hunt #60 (hunts/prime_zeta_rightmost/) we noticed that two published results, which do not cite each other, are the two directions of one statement, and that one paper's open conjecture is the other paper's theorem.
The conjecture, still printed as open. Belovas, Cepaityte and Sabaliauskas, "On the zero-free region and the distribution of zeros of the prime zeta function", An. St. Univ. Ovidius Constanta Ser. Mat. 33(2) (2025) 27-44, DOI 10.2478/auom-2025-0017 (open access). Their Theorem 1 states that the prime zeta function P(s) = sum_p p^(-s) has no zeros in the half-plane sigma > sigma_0, where sigma_0 = 1.77954465354699... is the root of U(sigma) = 2^(1-sigma) - P(sigma). Their Conjecture 1 states that lim_{T -> infinity} sigma_T = sigma_0, where sigma_T is the supremum of the real parts of the zeros of P with |t| < T. Their Remark 1 records sigma_M = 1.682628788045196... for |t| < 200000, so their numerics leave a gap of about 0.097 to the wall.
The theorem that settles it. Sepulcre and Vidal, "On the real projections of zeros of analytic almost periodic functions", Carpathian J. Math. 38 (2022) no. 2, 489-501 (preprint arXiv:1805.02041, 2018), Theorem 4.3 = preprint Theorem 6. For an almost periodic function f on a vertical strip U with Dirichlet series sum_{n >= 1} a_n e^{lambda_n s}, frequencies {lambda_n} Q-linearly independent, and sigma_0 in that strip, sigma_0 lies in the closure of the set of real projections of the zeros of f if and only if |a_j| e^{sigma_0 lambda_j} <= sum_{i != j} |a_i| e^{sigma_0 lambda_i} for every j.
The specialization. Take f = P on a fixed strip {1 + delta < Re s < infinity} with 0 < delta < sigma_0 - 1. Almost periodicity holds uniformly on that strip (P is bounded there); the frequencies -log p are Q-linearly independent by unique factorization; there are infinitely many terms; and the domination condition collapses, because p -> 2 p^(-sigma) is strictly decreasing so the index p = 2 binds, to
2 p_j^(-sigma_0) <= P(sigma_0) for all j <=> 2^(1-sigma_0) - P(sigma_0) <= 0,
which is exactly Belovas et al.'s U(sigma_0) <= 0. Hence:
- the "only if" half re-proves their Theorem 1 (no zeros with Re s > sigma_0);
- the "if" half gives closure{Re s : P(s) = 0} intersect (1, infinity) = (1, sigma_0], and since sigma_T is nondecreasing and bounded above by sigma_0, it forces sigma_T -> sigma_0. That is their Conjecture 1.
Provenance and grade. This is a literature observation, not a result of this repository: no new mathematics is involved, and both statements are used exactly as published. Grade: new as a connection between two published papers; the underlying theorems are theirs.
Where the working is. hunts/prime_zeta_rightmost/PRIOR-ART.md sections 6 and 7 carry the verbatim statements of both results, the hypothesis-by-hypothesis specialization, and a gap analysis of the bridge (two harmless technicalities noted: max versus sup on an open condition, and sigma_T being undefined for small T). Section 10 records why our own search missed both papers, which is the more transferable lesson. docs/30-prime-zeta-rightmost-zeros.md is the public reading-course page, rewritten to lead with the prior art.
What is not claimed. Nothing here bears on the Riemann Hypothesis: every zero discussed lies in Re s > 1 and belongs to P, not to zeta. Nothing here is kernel-checked. We have not contacted the authors and have made no attempt to publish; whether anyone pursues this is not what this issue is for.